Non-associative gauge theory
A talk in the Oberseminar Geometric Analysis series by
Sergey Grigorian from University of Texas Rio Grande Valley, Edinburg, TX, USA
| Abstract: | In this talk, we generalize some results from standard gauge theory to a non-associative setting. Non-associative gauge theory is based on smooth loops, which are the non-associative analogues of Lie groups. The main components of this theory include a finite-dimensional smooth loop, together with its tangent algebra and pseudoautomorphism group, and a smooth manifold with a principal bundle, with the structure group being the pseudoautomorphism group. A configuration in this theory is defined by a connection on the principal bundle and a loop-valued section of an associated bundle. Each configuration defines an associated quantity, known as the torsion, which is a tangent algebra-valued 1-form, and is another key object in the theory. Given a fixed connection, we prove existence of configurations with divergence-free torsion, given a sufficiently small torsion in a Sobolev norm. This result is analogous to the existence of the Coulomb gauge in standard gauge theory. We will then also show how these results apply to G2-geometry on 7-dimensional manifolds. Within the CRC this talk is associated to the project(s): A3 |